Download - MURI ADCN Workshop
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MURI ADCN Workshop
John Doyle, Steven LowEAS, CaltechOSU, Columbus
October 14, 2010
Post-docsLijun ChenKrister JacobssonNader MoteeChee-Wei Tan
Grad studentsMasoud FarivaJavad LavaeiJK NairSomayeh Sojoudi
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Outline Overview of Caltech projects (40 mins,
Low)
Optimal wireless protocols and devices (40 mins, Lavaei)
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Overview Heavy-tailed traffic File fragmentation to mitigate heavy-tailed
delay Tail-robust scheduling algorithms
Wireless Random access game Smart antenna design Power control
Congestion control Effect of ack-clocking Reverse-engineering transients
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Overview Heavy-tailed traffic File fragmentation to mitigate heavy-tailed
delay Tail-robust scheduling algorithms
Wireless Random access game Smart antenna design (Javad Lavaei) Power control
Congestion control Effect of ack-clocking Reverse-engineering transients
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File fragmentation
File fragmentation over an unreliable channelJ. Nair, M. Andreasson, L. Andrew, S. Low and J. Doyle. IEEE Infocom, San Diego, CA, March 2010
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File fragmentation: summary Motivation: how to mitigate heavy tail? Recent work showed file transfer time can be
heavy-tailed even if file size is light-tailed(Fiorini, Sheahan, & Lipsky, 2005; Jelenkovic & Tan 2007; etc.)
Results Independent or bounded fragmentation
preserves light-tailedness Constant fragmentation min expected delay Asymptotically optimal design: blind
fragmentation Optimal or blind fragmentation preserves tail
index
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Model Given file of random size L L is fragmented into K packets for
transmission at unit rate n-th transmission of size
n-th transmission is successful if
where are iid with distribution F
nxfile fragment constant overhead
nn xA
nA
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Model
LlxAxll nnnnn
1
1 )( 1
remaining file size at time n+1
fragment size at n
per-packet overhead
iid random var of distr F
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Model
LlxAxll nnnnn
1
1 )( 1
per-stage cost: )0()( nnn lx 1
total cost:
11
)0( )()(n
nnn
n lxLT 1
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Prior work
1
1
1
)0( )()(
)(
nnn
nnnnn
lxLT
LlxAxll
1
1
Theorem [Fiorini, Sheahan, & Lipsky, 2005; Jelenkovic & Tan 2007]
Without fragmentation T(L) has heavy tail even when L is light-tailed,
provided F has unbounded support
nLxn
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Result: LT-preserving frag
independent fragmentation: nnnn XlXx iid ,,min
boundedfragmentation:
TheoremWith independent frag or bounded frag:T(L) is light-tailed provided L is light-tailed
Then, heavy-tailed delay originates only from heavy-tailed files
nn lbxb ,min
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Result: optimal fragmentation
TheoremConstant fragmentation is uniquely optimal• Optimal #fragments: K*(L) =
• Optimal fragment size: x*(L) = L/K*(L)
•
per-bit cost:)(
)(
xFx
xxg
)(minarg0
xgax
aLinteger
LaaLx
Laa
/1)(
/1*
)( min LTxE
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Result: blind fragmentation
Theorem• for all L• Blind fragmentation is asymptotically
optimal
)(minarg0
xgax
LaLx as )(*
blind fragmentation: nn lax ,min
)()()( * aagLJLJ a
expected total cost: )(:)( LTLJ xx E
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Result: tail distribution of T(L)
Theorem• If L light-tailed, so is T(L) • If L RV(a) (heavy-tailed), so is T(L)
)(~)(
)(~)(*
agtLPtLTP
agtLPtLTP
a
optimal frag: )(,...,1 ),( ** LKnLxxn
blind frag: nn lax ,min
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Overview Heavy-tailed traffic File fragmentation to mitigate heavy-tailed
delay Tail-robust scheduling algorithms
Wireless Random access game Smart antenna design (Javad Lavaei) Power control
Congestion control Effect of ack-clocking Reverse-engineering transients
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Tail-robust scheduling
Tail-robust scheduling via Limited Processor SharingJ. Nair, A. Wierman, and B. Zwart.Proc. IFIP Performance, 2010; to appear in Performance Evaluation
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The “simplest” scheduling model
Q: What policy minimizes mean response time?A: Shortest Remaining Processing Time (SRPT)
RobustOptimal regardless of interarrival times, job sizes, etc.
A Wierman
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Q: Can a policy be optimal & robust for the tail?
Power-law job sizes Light-tailed job sizesWe’ll study the decay rate:
log P( )( ) limt
X tγ Xt
We’ll study the tail index:log ( )Γ( ) lim
logt
P X tXt
Lot’s of analysis over the last 20+ years…
Γ( )P( ) XX t t ( )P( ) γ X tX t e A Wierman
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SRPT Optimal [NWZ 08] Worst possible [NZ 06]Optimal [BBQZ 06] Worst possible [MZ 06]Worst possible [B76] Optimal [RS 01]Optimal [MT 80] Worst possible [NWZ 08]Worst possible [A99] Worst possible [N 07]
Power-law sizes Light-tailed sizes
Q: Can a policy be optimal & robust for the tail?
PSFCFSPLCFSLCFS
Lot’s of analysis over the last 20+ years…
A Wierman
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^(non-learning)
A: NO!Q: Can a policy be optimal & robust for the tail?
A Wierman
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Theorem: There does not exist a work-conserving,online, non-learning scheduling policy ν that has:
for all ε>0 and work-conserving, online policies πunder both light-tailed and power-law job sizes.
1( )limsup( )
εν
x π
P T xP T x
Corollary:Optimal under power-laws worst-case under light-tails,and vice-versa
A Wierman
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Q: Can a policy be optimal & robust for the tail? ^
(non-learning)
A: NO!
Q: Can a policy be weakly robust for the tail? ^
(non-learning)
better-than-worst-case under bothlight-tailed and power-law workloads
A: No known policies are.
A Wierman
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Our candidate: Limited Processor Sharing, LPS(c)
PSFCFS queue at most c jobs
is weakly robust and optimal for large classes of power-law and light-tailed distributions.
1 11
cρ
…but it uses ρ
A Wierman
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c=1FCFS
c=∞PS
Power-law
Light-tailed
Response time tail gets lighter
Response time tail gets lighter
c
c
A Wierman
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c=1FCFS
c=∞PS
Light-tailed
kc ≥ 2
better-than-worst-case
better-than-worst-case
Power-law
c < ∞
A Wierman
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c=1FCFS
c=∞PS
Light-tailed
better-than-worst-case
better-than-worst-case
optimal (if sizes have a finite variance)
optimal (if sizes are more “variable” thanan Exponential dist.)
1 11
cρ
Power-law
A Wierman
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Overview Heavy-tailed traffic File fragmentation to mitigate heavy-tailed
delay Tail-robust scheduling algorithms
Wireless Random access game Smart antenna design (Javad Lavaei) Power control
Congestion control Effect of ack-clocking Reverse-engineering transients
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Random access game
Random Access Game and MAC Design, L. Chen, S. H. Low and J. C. Doyle, IEEE/ACM Transactions on Networking, 2010
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Contention-based MAC (contention control)
Two components A contention resolution algorithm: adjusts channel
access probability in response to the contention A feedback mechanism: updates a contention
measure and sends it back to wireless nodes
)(tpi
)(tqi
L. Chen
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Dynamical model
The exact form of and are determined by or can be designed for the specific MAC protocol
Present a game-theoretic model to understand the dynamical system (1) and use it to design new protocols
)(tpi
)(tqi
))(()1())(),(()1(
tptqtqtptp
ii
iiii
GF
iF iG(1)
L. Chen
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Random access game
),( iiii qpp F
)( iii pFq
iiiii dppFpU )()(
fixed point
Only determined by the contention resolution algorithm Usually continuous, increasing and concave
))(()1())(),(()1(
tptqtqtptp
ii
iiii
GF
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Definition: A random access game is defined as a quadruple is a set of players (wireless nodes)
Strategy with
Payoff function with given contention measure
MAC (i.e., system (1)) as strategy update algorithm achieving the equilibrium of random access game The equilibrium properties can be understood and
designed through the specification of and The adaptation of channel access probability can
be specified through , corresponding to different strategies to approach the equilibrium.
})(,)( ,)( ,{ : NiiNiiNii quSN GN
]},[|{ : iiiii wppS 10 ii wv
)()( : )( pqppUpu iiiii (p)q ii G
G
iU iq
G)(F,
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Conditional collision probability as contention measure
Assumptions (single cell wireless LANs): A0: is continuously differentiable, strictly
concave, and with bounded curvature away from zero, i.e.,
A1: let and denote the smallest eigenvalue of by . Then, .
A2: functions are all strictly increasing or all strictly decreasing
)1(1 jIji p(p)qi
)(iU
0/1)(/1/1 '' ii pU)1()( ii
pp )(2 p min 0min
))(1)(1()( 'iiiii pUpp
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Equilibrium Theorem: Under assumption A0, there exists a
Nash equilibrium for random access game. Suppose additionally A1 holds. Then random access game has a unique Nash equilibrium. A channel access probability is a Nash
equilibrium of random access game, if
Proof: By showing the equilibrium condition
is the optimality condition for a strictly convex optimization problem.
*p
. , ,),(),( *1
** NiSpppuppu iiiiiii
iiiiiii SppppqpU ,0)))(()(( ***'
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Nontrivial Nash Equilibrium A Nash equilibrium is a nontrivial equilibrium if
for all nodes , the equilibrium strategy satisfies
and trivial equilibrium otherwise. Theorem: Suppose A2 holds. If the random
access game has a nontrivial Nash equilibrium, it must be unique. Proof by contradiction: Note that a nontrivial Nash
equilibrium
*ipi
),()( **' pqpU iii
., ),()()( *** Njippp jjii
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Definition: A Nash equilibrium is said to be symmetric if for all , and an asymmetric equilibrium otherwise. By symmetry, there must have multiple
asymmetric equilibria if there exists any. Theorem: For a system with several classes of
users, suppose A1 and A2 hold. If random access game has a nontrivial equilibrium, it must be unique and symmetric. Guarantees fair sharing of wireless channel among the same
class of wireless nodes Provides service differentiation among different
classes of wireless nodes
*p**ji pp Nji ,
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Gradient play
Have a nice economic interpretation Theorem: Suppose A0 and A1 hold. The
gradient play converges to the unique Nash equilibrium of the random access game if for any , the stepsize
Proof by Lyapunov method. Also studied its robust verification to the
estimation error.
isiiiiii tpqtpUttptp )))](())(()(()([)1( '
.1||
2
N
fi
i
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A concrete MAC design Consider a single-cell network with classes
of users Each class associated with a weight
Assume Want to achieve maximal throughput under
the weighted fairness constraint
L
l
.,1 , LmlTT
m
l
m
l
l.2max1 L
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Utility design Let . Under the assumption of Poisson
arrival, the throughput achieves maximum at that satisfies
the duration of idle slot, the duration of a collision
Under the decoupling approximation, to achieve weighted fairness requires
ii
p
*
ce T/1)1(**
cT
.,1 , Lmlpp
m
l
m
l
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Requires
A convenient choice
Utility function
*
)(
)())(1)(1()(
*
'
eph
phpUpp
l
l
l
llllll
)1()(*
l
l
l
l peph
],0[
)1ln()11()1()(*
*
wp
pepepU
l
ll
ll
ll
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Equilibrium and dynamics Theorem: Suppose
The random access game has a unique and nontrivial equilibrium
The gradient play converges if the stepsize
./11
1/1
1
maxmax
*
*
*
ewe
e
.1||
2
N
fi
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Performance: throughput
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Performance: collision
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Performance: short-term fairness
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Performance: dynamic scenario
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Performance: service differentiation
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A natural progression
Centralized optimization
Distributed but cooperative actions with rich information and signaling allowed
Less cooperation(economic perspective)
Less information or signaling available(engineering perspective)
Optimization
Game theory
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Overview Heavy-tailed traffic File fragmentation to mitigate heavy-tailed
delay Tail-robust scheduling algorithms
Wireless Random access game Smart antenna design (Javad Lavaei) Power control
Congestion control Effect of ack-clocking Reverse-engineering transients